General property of Fitting series

Question

Let $G$ be a finite solvable group. $F(G)$ (Fitting subgroup) is defined to largest normal nilpotent group contained in $G$.

Then $F_2(G)$ is defined to be inverse image of $F(G/F(G))$. i.e $F_2(G)/F(G)$ is Fitting subgroup of $G/F(G)$.

Inductivly,we have $F_k(G)$. We have this series;

$$1<F(G)<F_2(G)<....<F_{k-1}(G)<G=F_k(G)$$

Then $k$ is defined as Fittling lenght of $G$ and denoted by $f(G)=k$.

If $N$ is a normal subgroup of $G$, It is easy to see that $F_k(G)\cap N=F_k(N)$.

And in general, $f(G)\leq f(N)+f(G/N)$.

I wonder following questions;

• Let $N$ be a normal subgroup of $G$ such that $f(N)=f(G)$. Then, $G/N$ is nilpotent. (True or not)
• Let $M,N$ are two normal subgroup of $G$. In which condition does following hold ? $f(G)-f(N)=f(G/N)-f(M/N)$

Any source suggestion is welcome. Is there any good source examining basic properties of Fitting length ? or asnwering some basic question like this. Any direct answer to these question is also welcome.

Answer 1

The answer to the first question is no. Let $H$ be any group of Fitting length $n > 1$ and $G = H \times H$. Then, taking $N = H \times \{1\}$, $f(N)=f(G)=n$, but $G/N \cong N$ has Fitting length $n$, so it is not nilpotent.